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Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:S\to S$ with $LG=1$. Retarded means that for every $g\in S$ whose support is disjoint from the past causal cone of $x$, the support of the retarded solution $g=Gf$ of $Lf=g$ is disjoint from the past causal cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

Which classes of (scalar or systems of) linear first or second order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:S\to S$ with $LG=1$. Retarded means that for every $g\in S$ whose support is disjoint from the past causal cone of $x$, the support of the retarded solution $g=Gf$ of $Lf=g$ is disjoint from the past causal cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

A particular case I am interested in is hyperbolic $L$ of the form $$L=\pmatrix{\alpha & (a-d)^* \cr b-d & B},$$ where $d$ is the exterior derivative, ${}^*$ is the Minkowski adjoint, $\alpha$ is a scalar field on Minkowski space, $a,b$ are covector fields, and $B$ is a matrix field mapping covectors to covectors. Hyperbolicity is defined as in Wikipedia.

Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that for every $g\in S$ whose support is disjoint from some past causal cone, there is a unique retarded solution $f\in S$ of $Lf=g$, i.e., such that the support of $g$ is disjoint from the past cone of $x$ for every $x$ such that the support of $g$ is disjoint from the past cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that there is a retarded Green's function $G:S\to S$ with $LG=1$. Retarded means that for every $g\in S$ whose support is disjoint from the past causal cone of $x$, the support of the retarded solution $g=Gf$ of $Lf=g$ is disjoint from the past causal cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that for every $g\in S$, there is a unique retarded solution $f\in S$ of $Lf=g$? When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?

Which classes of (scalar or systems of) linear second-order hyperbolic PDEs $Lf=g$ in $n$ variables and spaces $S$ of functions have the property that for every $g\in S$ whose support is disjoint from some past causal cone, there is a unique retarded solution $f\in S$ of $Lf=g$, i.e., such that the support of $g$ is disjoint from the past cone of $x$ for every $x$ such that the support of $g$ is disjoint from the past cone of $x$.

When can one choose $S$ as a space of smooth (i.e., $C^\infty$) functions?

Where can I read about the mathematical tools for studying this and similar questions?